Journal of the Textile Institute Proceedings and Abstracts
ISSN: 0368-4504 (Print) (Online) Journal homepage: http://www.tandfonline.com/loi/tjti19
Some Physical Quantities of Mule Yarns
A. E. Oxley M.A., D.Sc., F. Inst. P. & F. T. Peirce B.Sc., A. Inst. P
To cite this article: A. E. Oxley M.A., D.Sc., F. Inst. P. & F. T. Peirce B.Sc., A. Inst. P (1922) Some Physical Quantities of Mule Yarns, Journal of the Textile Institute Proceedings and Abstracts, 13:9, 172-188, DOI: 10.1080/03684504.1922.11673741
To link to this article: http://dx.doi.org/10.1080/03684504.1922.11673741
Published online: 22 Feb 2016.
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Download by: [University of Cincinnati Libraries] Date: 21 March 2016, At: 21:32 SOME PHYSICAL QUANTITIES OF MULE YARNS.
By A. E. OxLEY, M.A., D.Sc., F. Inst. P., and F. T. PEIRCE, B.Sc., A. Inst. P
(The British Cotton Industry Research Association).
(1) THE RELATION BETWEEN THE TWIST AND THE AMOUNT OF FIBRE.
In a previous paper (this vol., pp.M-98) a method of measuring the regularity of a yarn was described, the quantity measmed being the thick ness under compression and small tension. This quantity, which might be termed the " hardness " is a function of the t\vist and number and fine ness of the fibres in a cross section. A large amount of yarn from various cops was examined by this method, the same lengths being afterwards tested in li'' pieces for twist. Thousands of readings were also taken of the breaking load of the same yarns, the test pieces being consecutive lengths of 31 inches. The twist specimens were marked and preserved in the order of testing so that they could be oxamined to find the relation between the amount of raw material and the twist at the same portion of the yarn, since these two factors should determine completely the qualities of the yarn, given the same quality of raw material. The fibres were counted under the microscope with a fairly low power, the ends of the specimens being care fully teased out with a needle. Four yarns of different counts were examined, some hundreds of consecutive measurements being made on each. The number of fibres per section was next plotted against the distance along the yarn. This number varied by amounts up to about 50% above and below the mean, the variations showing no trace of periodicity, so far as can be traced by the eye. The regular periodicity of hardness, twist and tensile strength must be a pure spinning effect introduced by the transmission of all the twist from the spindle tip. (1-oc. cit. p.57.)
Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 Many variable quantities enter into each of the hundreds of observa tions of fibre number and corresponding twist. The effect of the in dependent variable, the quantity of fibre, is determined not only by the number of fibres, but also by the shape and size of the cross-sections, the lengths and convolutions of the hairs, and the properties of the cuticle, etc. Superimposed on this effect is the periodic spinning effect previously refeNed to. The large number of readings, however, enables unknown random variations to be eliminated, and provides a means of isolating the effect on the twist of the two known quantities-(1) the number of fibres at a given section, and (2) the distance of this section along the draw. Throughout the length of the same yarn, the only unknown variable which seriously affects the question is the average cross-section of the fibres at the point examined. As the twisting couple which determines the conditions of equilibrium depends on the square of the cross-sectional area, SOME PHYSICAL QUANTITIES OF MULE YARNS--DXLEY AND PEIRCE 173
any variation of this yuantity will have a large effect. In couniing the fibres, it was plainly evident that a statistical average was not arrived at !or individual sedions, bunches of fine fibres being frequently observed.
PLATE I. I I I ~ 60 ~ ~ 50 ~ ~ ;: ...::, ~ \ '•.. 40·,..~' ( .._; : ., I ;:l 0 so ()"' ~ .-n 0 ~""" 0 20
...,; 4 1 N 3"10 I I 0 I l.'\ ()"' ] ~I I !i r.· .I II ·~ I ;~\ :• I...... • i \'\ ll 1.4 I ~ I I 0 I ~ ~ .._; ~ s:l ~ 40 0 ()"' ~ _-n 0 I 0 N ~ ~ 0 1.() ~ z0
Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 0 ()"' .. I tii 1 ~"I i •I j. ~·.. . A, u I I·' I I I ~ 4 \ IJ I .. I '10 ...... "" "'' ' tl I 0 10 20 30 40 50 60 70 TE;:;r Ptec>EB s tf; cl?M. Horizontal distance between consecutive points= H" of actual yarn.
The results already obtained for the twists per inch, in consecutive specimens, were plotted alongside the corresponding values of the fibre number. The two graphs showed an almost invariable correspondence, the 174 SOME PHYSICAL QUANTITIES OF MULE YARN5-0XLEY AND PEIRCE
variations being in oppot:;ite ways. There is, however, no regularit~ in the amount of the v'ariations for individual sections, and many exceptiOns occur even in direction, owing to the periodic variation along the draw and the random variations in fineness of the fibres. Two representative srJecimens of s:uch graphs for lOO's and 200's counts are shown in Plate I. Though these effects vail:y appreciably for individual seetions, they may be sufficiently eliminated to find a simple relation between fibre number and twist by finding the average twist whieh corresponds to given fibre numbers taken over the whole number of specimens from the same yarn. In this way, an average value is obtained which indudes the influence of up to 800 fibres and this substantially eliminates the variation. The effect of the periodic variation is also distributed at .random, and so the relation between twist and fibre number is obtained as an average over the length of the draw. The first yarn examined was a cop of 200's count, t\ea Island, the fibre numbers in 400 sections having been first counted. The average values of the various quantities involved are given in Table I, and the average value of the twists in ll'elation to the corresponding fibre number is given in Table II. These average twists were plotted against the number of fibres and a smooth curve drawn through the points. This curve was found to be expressed by an equation of the form (n- a). t1 = c...... (l).
where n is the number of fibres in a section of a specimen having t twists per inch, and a P.nd c are two constants for the yarn. The values of n were also plotted against 1Jt2, giving a straight line from which the values of a and c could be more easily estimated. Many other types of equation were tried, but the above was found to be the best for all the yarns, individually and collectively. For instance, an equation of the form n.tb=c may be found, which expresses the results as well as (1) over the range involved, but a different incommensurable value of the power b is necessary for the different yarns. The best value of a was found from the graph connecting n and ljt2 to the nearest integer, and then c was adjusted to satisfy the points nearest the mean, i.e., those representing the largest number of sections and the truest average. This determination is as exact as possible, for, if c be adjusted to small va.riations of a, the difference is only appreciable at the extremities of the curve where the points are not very determinate. The same methods were then applied to the other yarns, which were also found to follow this law with an exactitude depending on the number of readings used and on the number and uniformity of the fibres. ThE 32's yarn gave particular trouble in counting the fibres owing to the large Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 number in each section. For this reason, to get a better average, the fibre numbers are grouped in threes, the number given being the middle one. The resulting curves are shown on Plates IIa, lib, lie and lid. In the following lists of particulars, the observed counts (represented by N) were found from the unused portion of the cops, the actual yarn previously used not having been weighed. These values are taken in calculations involving the counts as somewhat more probable than the counts to which the yarn was said to be spun. The quantity p is the constant in the mill-rule, viz., Twists per inch= constant x vcounts. For this type of yarn, it is usually taken as 3·2 approximately and the change wheels adjusted accordingly. It will be s.een that there is a difference (loss) due to fib.re slip, of about 20% between the twists put in at the spindle and that observed in the yarn (l·oc. cit. p.93). Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016
TABLE II. COP No. 50. 200's COUNT. SEA ISLAND. FIBRE NUMBERS. 1>2 13 14 15 1& 11 18 19 20 21 22 23 24 25 26 27 28 2!1 30 31 32 33 34 35 36 37 38 39 40 50 51 51 H 38 32 42 43 36 41 41 25 32 27 21 35 27 32 :II 26 30 20 23 lfo: 35 U> Ill 51 39 4S 34 39 4.::; 3S 35 30 2S 40 :lO 27 33 26 30 2~ 23 211 2fi 20 '27 '!.~ 2:1 0 ~ 50 60 48 4fi 41 37 37 33 31 31 40 29 21 35 42 28 24 36 41 40 34 :.15 :.:a ['1 64 40 28 55 37 25 36 32 33 27 31 30 31 29 24 32 35 29 32 Hi ., 42 37 58 36 33 45 31 33 30 26 39 22 26 42 42 34 27 30 42 35 25 54 30 35 24 29 25 30 31 29 36 25 22 41 ~:! 22 u;~ 36 42 32 38 28 til 27 27 25 42 34 28 35 ~7 34 32 18 37 45 2H 4S 53 as 41 32 32 47 35 42 31 4o :II 22 25 '27 n 34 49 46 28 49 42 34 43 44 27 23 41 32 30 30 36 r> 46 49 31 35 SG 30 35 35 30 35 3S 39 28 30 25 25 0 48 38 44 35 31 42 44 45 30 30 33 41 24 22 34 c 62 35 30 39 5;< :16 38 32 2!! 38 33 2li 34 33 20 :1> 52 66 30 4f, 40 47 37 :l7 41 34 4S 24 28 23 z 40 55 40 34 35 41 36 40 30 35 28 33 32 -1 3-1 so 43 29 48 40 30 38 47 34 29 21 26 ::j 38 4:1 47 4~ 54 47 26 24 30 20 32 39 39 37 28 25 40 41 37 34 20 21 ~ 40 25 38 30 36 29 33 37 31 31 19 0 35 48 38 40 42 31 30 40 35 22 19 .., 37 34 39 28 33 30 42 44 30 :u ~ 33 37 52 42 24 41 36 25 29 c 38 ~3 :lll 42 27 37 28 32 26 49 :18 42 40 :19 28 40 46 !;-; 37 39 20 ~9 27 29 35 ><: 49 32 33 36 29 >::c 30 35 40 37 34 z 47 4~ :m 37 31 u; 36 2~ 2-1 41 31 I 41 39 49 ~6 22 0 37 27 26 fS 38 34 40 31 33 31 ~ 39 30 30 ?l 30 31 28 t:i 22 25 20 25 ['1""" 34 :;;; 35 0 29 ['1 161 .... 1132 202 298 616 955 857 1279 14S8 1013 84:1 782 1173 9-16 567 592 411 4~9 368 27H 238 91 50 .... 42 Ill 3 3 4 7 IS 23 22 35 39 29 2-1 24 35 29 19 20 13 15 12 10 8 4 2 ...... 2 3 53·6 54·0 50·5 4:.!·6 41-1 41·5 39·0 36·S 37·-l 35·0 35·1 32·6 33·5 32·B 29·8 29•6 31·6 29·9 30·7 27·6 29·8 22·8 25 ...... ~1·0 27·0 Total Twists ...... 13910 No. of Spec.imros .... 400 AYerage .... 34·~ ::::; "' 176 SOME PHYSICAL QUANTITIES OF MULE YARN5-0XLEY AND PEIRCE
PLATE IIa. Cop No. 50, 200's Sea Island. (n-5) .t2=22,000. 60
!55
!50
r45 40 :§: .rl (J 35 .E .,... !50 ~ t; 25 -~ X .... X .,20 40 bO "'... ! !::; < ::t 20 10 :!! 5 ~ 10
10 40
PLATE Ilb. Cop No. 19, 175's Sea Island. (n-7) .t2= 13,000. 6~
60
55
50
f 45 ... '10 ~ I Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 ·-...., 35 ....~30 .~ :t 25 ....., .,..~20 10 > 15 ~ 30 < ~
10 ~ 20 ~ 5 ~ 10
0 I I (I ...~cduJJ\111.\\iil 6 e 10 17 14 .~ 18 20 r.::: 2'4 26 28 30 32 34 Number of fibres per cross section (n) --7 PLATE lie. CJp No. 23, tOO's Egyptian. (11-11) .t2= 13,i00. co 10 X \() 10 X 10"'" )( C\1 )t 10 X 0 )( 10
)( ~ )(
\() X ot X I I ot X I I ot I X I / X / ,: ~ l I I I ~.§ ....C) QJ 1/J ~~... C) .O"' J I() g. rn QJ I 0 Cll !!:! ~ ~ ~ ...0 ~ 10 ~ 5! Ill ~ t\1 2 ~ Q 1.1) 0 Average twist per inch (t) ---? 178 SOME PHYSICAL QUANTITIES OF MULE YARN5-0XLEY AND PEIRct Tho quantity m 11 is the average mass per fibre per yard of yarn, whil'h, corrected for oblitjuity of fibres (p.l'i8), is a measure of the fineness of the cotton. If n is the mean number of fibres per cross-section, I mo = 840 Nn. The length of the period l is found by inspedion of the graph of the twists and the quantity (n-a) .t2 and agrees well with .that found from the photo graphs, from the tensile tests, and from the actual length of the mule draw (Zoe. cit. pp.5!J, 72). Pr..ATE IId. Cop No. 48, 32's Egyptian. (n-44) .t2:5,560. .. u .. r 20 ~ • -5 16 .!3 ... 4.1... Q, 1il -~ ... oC) 4.1 bO a fl 4.1 :;. < 0 4 20 I ·· .... ···'~ e. ll .. 0 ~--.(tt .. rJ-:·--r-+--b----- • 11:5 '40 .., 60 ~ ~ ~ ~ 00 M ~ ~ 00 ~ ~ ~ ~ ~ Number of fibres per cross section (n' -~ In the lOO's yarn, two different portions were examined. These nre recorded separately to show the probable variation in the quantities esti mated in this way. TABLE 1.-PARTICULI\RS OF THE YARNS EXAMINED. Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 COUNTS 200's, OBSERVED (N) 197's. SE·A ISLAND CoTTON. Cop 50. Photographic test 48 (see Plate V. in paper referred to above). Number of Sections 400. • Mean number of fibres per section (n) = 23.7. t Sub-mean (n') = 19·9. Irregularity, measured by the quantity n - n' = 16 %· - n Mean number of twists per inch (tj = 34·8. . l P = vN = 2·48. 1 m0 = 2·57 X 10·• lbs. per yard. • Throughout the paper, the quantities marked with a bar (n) are averages over a length of yarn; the plain symbol (n) representing the value for an individual section. t The Sub-mean is the mean of all those observations which are equal to or less than the mean of all the observations. It is an indication of the extent of variation below the mean which has an important practical significance. SOME PHYSICAL QUANTITIES OF MULE YARNS-OXLEY AND PEIRCE 179 The best equations connecting the number of fibres, n, and the corresponding twists per inch, t, for individual sections are :- (n- 5). t• = 22,000, and (ii- 5). l' = 22,600, which shows the closeness of the" tit" when mean values of n and t are substituted. The length of the period, l, = 55 inches. Photographic test No. 48 gave 56 inches. CoUNTS 175's. SEA ISLAND CoTTON. Cop 19. Photographic tests Nos. 50 and 51. (Plates III. and IV. in paper refe1red to above). Number of sections 357. 11 = 23·6 fibres per section, 11' = 19·4. Irregularity = 17·8 % f = 29·5 twists per inch. p = 2·23. m. = 2·89 x lQ-7 lbs. per yard. Best equations :-(n- 7). tt = 13,000, and (fl- 7). ft = 14,350. l = 60 inches. Photographic test No. 51 gave 60 inches. CoUNTs tOO's. N = 103's. EGYPTIAN COTTON. Cop 23. Two series (a) and (b). 629 sections. (a) Photographic test No. 40, 300 sections; (b) Photographic tests Nos. 42 and 43, (Plates I. and II. in paper referred to above), 329 sections. Ia) 11 = 34.4, n' = 29·0. Irregularity 15·7 %· (b) fl = 36·2, n' = 30·7. Irregularity 15·3 %. (a) i = 25·0 twists per inch. p = 2·46. (b) I = 24·3 twists per inch. p = 2·40. 7 (a) m 0 = 3·36 x 10· lbs. per yard. (b) "'• = 3·20 x I0-7 lbs. per yard. I = 62·5 inches. The photographic tests showed that the period varied from 60·9 to 63·~ inches; the mean of thcsl! is 62·3 inches. Best equations :- (n - 11). t• = 13,700, and (a) (fJ- 11). I' = 14,320; (b) (H- 11). 12 = 14,880. COUNTS 3:.!'s. N = 33's. EGYPTIAN COTTON. Cop 4!:l. 257 sections. n = 83·6, n' = 65·5. Irregularity 21·6 %. t = 13·0 twists per inch. p = 2·26. 7 m0 = 4·32 X· I0- lbs. per yard. t = 58 inches. Photographic test No. 46 (Plate VII. in paper referred to above) gave 58 inches. _ Best equations :-(n- 44). t• = 5,560, and (n- 44). t• = 6,700. (II.) 1HE FIBRE CROSS-SECfiON. The frequency curves of the fibre numbers (Plates lin, b, c, d) are of interest, as they show, espe<·inlly in the finer conntfl, an C'xtm '' mode " or maximum above the mean. This is undoubtedly Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 distribution than the 32 ·R. This is probably due to the fad that they were combed, and it would be expected that the opposite would be true if all the cottons had been similarly treated. A measure of the fineness of the cotton may be obtAined from m 0 by applying a correction for the obliquity of the fibres. If s is the normal cross-section of a fibre, s 0 the cross-section normal to the axis of the yarn, s = S 0 • cos 8, where 8 is the angle between the fibre and the axis. Assum ing a constant density of fibres over the cross-section, _s_ =-1_, J.R 27t1'. dr. cos 6 = .YI+4,;•Rtt•- 1 s. ,;R' o 2,;•R•t•. 1 since "'u=tan- 127tri=cos-t v''4"",;""•=;,,;=.,,;=+;=;=1=- This is. a small corre~tion .and. in estimating its value, it may be said that approximately t2 vanes d1reetly and R2 indirectly as the counts. Hence 180 SOME PHYSICAL QUANTITIES OF MULE YARN5-0XU:Y AND PEIHCE R2t2 is constant, and, from measured diameters (see below, p. IRI)) and known twists, its average value for tho different yarns is 1/500 and 4rr2R2t2=0·079. Therefore (s 0-s) fs 0 =0·02, i.e., a correction of 2% should be subtracted from t.he value obtained for 111 0 • Assuming the density of the fibre material to be 1·5, the cross-section (s), in square inches, and tho mass per yard in lbs. ( 1T1), are connected by the relation s= iii __1_ 49 __!_ ___1 __ 5·96 x to-a s . ins. . 1·95-840Ntr" 5ll" HI5-1680Nn- Ntl q This indicates a method of some general value, as it gives the area of cross-section of the walls of the fibre, averaged over thommnds of fibres. 'l'he actual numerical values, in this case, are mwertain to the ext{'nt of the probable variation of the counts because the aetual matlorinl usc1l had not been weighed. This quantity is the best one for defining the fineness of the cotton and has much more meaning than the visual diameter, or ribbon width and thickness, or even the cross-section obtained by direct measurement. With an irregular form such as that of the cotton fibre, diameter and area of cross-section are indefinite quantities unless the latter be tlefined as the volume per unit length, which is the quantity obtained above. The cross-sections found in this way are:- SEA ISLAND COTTO:<. EGYPTIAN (OTTON. Counts .... 200's. .... 175's. .... lOO's. .... 32's. s x 107 sq. in...... 1·28 .... 1·44 .... 1·63 2·15 'l'he variation shown indicates that the average diameter of the coarser Egyptian cotton is 1·3 tiuws that of the finer Sea Island. (III.) THE GENERALISED FORMULA OF TWIST AND FIBRE NUMBER. It has been shown, by the foregoing observations, that the. relat~on between the twists per inch and the number of fibres mny be generally expressed in the form (n-a) .t2=c where a and c are constants for any one yarn. This must express the conditions of equilibrium, as far as n and t are concerned, in the newly-spun yarn, i.e., the couple tending to untwist the yarn must be proportional to c. The couple also depends upon the clastic properties, and on the size and shape of the fibres. · 'l'o make this oxpression vali.l, on'n for the simplest ease, viz., a strckh of single yarn, the effel"t of varying eross-sedion of the fibres must be include Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 noticeable where the twist is excessive) as soon m; symmetry i,; clestroye ln this form the equation takes aceount of all the significant factors and is dimenRionally complete. Written in the form:- 2 s2(n- (IV.) THE MILL RULE. TWISTS PER INCH=CONST. x y'COUNTS. The above rule expressed in the symbols adopted in this paper is:- t=p.y'N or i.e. t2.840.n.m=P'· --- p• or s.n.t'=T680 If both sides of this expression are multiplied by (ii-a) .8 then ii. s. 2 (n - a) .I•= L. (n - al ..s. 1680 n It is a curious fact that the value of (ii-a) .8, despite the large variations n in the values of ii, 8, and a, for counts, 200's, 175's, lOO's, and 32's, is eonstant for all the yarns, to within the limit of error in determining these quantities, the greatest variation from the mean value, 1·05 x lQ--7 sq. in., being under 8%. This quantity might be termed the average effective cross-Rer.tion of a fibre. Inserting this value, the mill rule becomes - P" X 10-1 - 2 (- ) i• s2(n-a).t2 =-----rsoo Or S n;.a. -Q·625xlQ-lD ...... (4) From Table III, it is seen that s2.cfp2 is constant and equal to O·U2 x lQ-10. The greatest variation (7-i%) from the mean was found in the lOO's counts (cop 23), and even this would be reduced to 3% if a were given the value 12 instead of 11. It was difficult to decide which of these numbers gave the better fit. The quantities on which the value of the expression depends, viz., N, ii and a, vary from yarn to yarn by factors of 6, 4, and 9, so that the agreement is very decided. TABLE III. Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 No. of Cop .... so 19 23 48 Variety of Cotton .... Sea Island Sea Island. Egyptian. Egyptian . (a) 300 400 357 629 No. of Specimens examined (b) 329 257. Counts (mill) 200's. !75's. !OO's. 32'~. Counts (N. obs.) !97's. 103's. 33's. (a) 34·4 23·6 35·3 8:l-6 :n .... 23·7 (b) 36·2 (a) 25·0 34·8 29·5 24·6 13·0 / .... (bj 24·3 a .... 5 7 11 44 c.... 22,000 13,000 13,700 5,560 (a) 2·46 .. p 2·48 2·23 2·43 2·25 (b) 2-~0 " p = l/y'N. 182 SOME PHYSICAL QUANTITIES OF MULE YARNS-QXLEY AND PEIRCE TABLE II I. -continued, No. of Cop .... 5U 19 23 48 Variety of Cotton .... Sea Island s~a Island. Egyptian. Egyptian . 1 68 x 107 sq. in .... 1·28 1·63 (a) ' 2·16 t s 1·44 (b) 1·59 (tl- a).sx 101 1-01 1·01 (a) 1-14 - n 1-12 (b) 1-1 1 1·06. mean 1·0;, ~· C X }QlO 3·60 2·69 3·65 2·59 ~ s• c X }QlO ·599 ·604 670 .... ·608. mean·62 ~-p·-- T (ozs.) .... 1·47 1-48 2·80 5·63 Q='T.N_. p. 118·5 116 115 80 Breaking stress X 10-' 1·46 1·70 1·05 (lbs. per sq. in.) Diameter (ins.) i5 X 108 3·01 3·78 7·27 Thickness under compression in regularity tester, H < to• 0·9 1·5 3·4 (a) and (b) refer to different. series of tests. - 1 ~ ~-a 5 t = 16SONia t p2 =168d"Nn• Tho results obtained may be summarised as follows:- Obsorvations of twists aml fibre numbers lead to the expression (n-a) .t2 =c along the length of any one ~·m·n. This relation when generalised by statistical considerations becomes: s2 (n - a}t 2 =~ b.A.q. which holds for any yarn. The mill rule for spinning is shown to involve the equation s2 (n- a).t2 =P' x 0·62 X IQ-10 Thus this purely empirical rule is the simple.<;t possible expresRion, in terms of the average vr,lnes, whil'h nre nll that concern the spinner, of the rela tion between the amount of fibre and the twist. In spinning to the same p for all counts, a spinner obtains tho same value of M fb, a quantity which might be termecl the spinning force. In general M i b=p1 x0·62x10·10A.q. where P= .YN. It must here be noted that the value of p i;; not iJentical with the constant by vvhieh the mule gearing is adjusted but about 20% lower owing to fibre slip in spinning (l·oc. cit., p.93). The quantity M fb, a force varying only 'vith p2, suggestR itself as a Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 standard of conditioning, the residual value in a conclitioned yarn being a measure of the amount of conditioning. 'l'he absolute values of M and b and their variation with the counts can only be determined by measuring the couple in newly spun yarns. This is now being investigatecl. (V.) 1HE PERIODIC VARIATION OF TWIST IN MULE YARNS. It has been shown in the paper referrc>d to above (Plates I-XXYIII) that in mule yarns the twist, as well as the hardness and tensile strength. has a periodic maximum inclependent of the variation of th£l amount of fibre. This superimposed effect is greatest at those points of the yarn which were at the spindle tip during spinning. All the twist that the yarn acquires is fed in at this point. About 20% of this is lost through fibre slip, the remainder being diRtributecl throughout the clrnw, the trans mission of twist taking place in virtue of the elastic forces exerted by th·~ SOME PHYSICAL QUANTITIES OF MULE YARN5-0XLEY AND PEIRCE 183 fibre~. If the fibres were pedeetly elnstie and did not slip, equilibrium along eaeh draw >Yould be deterwined by a e011Htant couple and an even distribution of twist. Each transmitting element of the yarn, however, must be given a higher twist than it finaliy retains, in order to pass it on. Owing to imperfect elastic restitution, eaeh clement exerts a forec in un twi~ting \\·hieh is h.!ss than that by whieh it resisted torsion, so that, for e s•(n- a).t• =C0 j (d)...... (5). where C0 is the value of the twist at the rollers and d the distance along the draw measured from the rollers. Having now determined the relation of twist to the amount of fibre, it is possible to eliminate the effect of variations of twist due to irregularity of roving and to determine the true amount of variation of twist introduced by the mule meehanism. The values of the function (n-a). t2 were plotted, for the individual points, alongside the corresponding graphs for n and t (see Plate I). The gmph showed the gradual rise of twist and the maxi mum of the mule draw, but the large unknown variation of s2 caused irregularities. It is possible to eliminate this variation to a considerable extent by averaging the values for corresponding points of the periods. 'rhe length of the periods is known from the photographic tests, the tensile tests, and by inspection of the maxima shown in the graphs of t and c. This enabled the graphs to be cut up into an equal number of periods. To get a better average, the specimens were taken two at a time, i.e., over lengths of 2~in. for each point of the composite curve. This is suffi cient, not indeed to give a perfectly even curve, but to smooth out chance irregularities of fibre so that the true shape of the curve may be definitely seen. In Plate III, the resulting composite periods are shown for the quantities n, t, and 82.c, meaned over the specimens used in the foregoing observations. The smooth curves, in all cases, represent the values which should be given by averaging over an infinite number of periods. In parti('ular, the smooth curves for the twists are calculated from those of 82.(n-a).t2, by Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 dividing by the average value of 82.(11-a) thus eliminating the effect of varying fibre numbers. The statistieal method tends to eliminate (1) the variations in fibre cross-section, (2) the variation of number of fibres per cross-section. Thus it will be seen that the variation of twist with fibre number is much more consistent when treated in this way than it is fn any individual period. The number of fibre;;; still varies, though much less than in· the original periods. The irregularities remaining indicate the degree to which this limited amount of averaging falls short of statistical perfeetion which would give a straight line. The smooth r·nrves represent the values of twist, ete., corresponding to this straight line, i.e., to the icleal distribution of fibres, and therefore they represent the pure spinning effect. The curves have been arranged to show at a glance how the quantities concerned vary with eaeh other along the draw of a purtieulur yam and 184 SOME PHYSICAL QUANTITIES OF MULE YARNS-QXLEY AND PEIRCE how they vary from yarn to yarn. The twist anrl fibre numbers are on the sHme seale for each yarn, exeept for the 02\;, where the twi~;h; arc multi plied, and the fibre numbers divided by three, to show the variation in the same proportions as in the other yarns. The [·.urves for §2. (11-a). t 2 represent the function f (d) of et1uation (5), which holds for C,J(d), 1'vllb, t2 (with uniforlllity of fibres and their distri bution), or c in the original formula (1). 'fhe value of s2.c. i~; remarkably constant, varying less from ;yarn to yarn than along any one draw. If p, the spinning eonstant, on whieh only it dcpenrls, be taken as tho zero, the curves arc praetieally continuous. (See Plate III.) Taken together with the eurvel4 of Plate II, all the varying faetors in a yarn. are thus speeified and eorrelated. In the latter, the periorls are averaged out, in the former the variations of the fibre number, in both the unknown effeds of variation of fibre, ero~;s-seetion, etc:. These c:onulusions are inc:orporated in the eomplete formulw:- s2 (n- a).1 2 =" .j(d). and s2(n- a).i2=0·62x 10-top•=b-Allf. 0 q. It iK now po~sible to relate the praetieal qualities of a yarn to these quan tities on whieh they depend ( 1) along a given yarn, and (2) from yarn to yarn. The method of eompoHite periods luu; been applit•cl to the tests on ht·8nking load made on the san1P l'nps. A greater number of periods wa::; available for thil4 purpose nn Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 on lOO's yarn by varying the twist in the ;yarn. They fall in that part of the curve where the relationship is praetieally linear, even for the regions of high twist. This, of [·ourse, in a weft yarn; in yarns of higher twist, the part of the eurve represented may attain and reach over the point of maximum strength (See " Regularity of Single Yarns," Zoe. cit., p.90). Over the general range of twist, we can assume very approximately that, other things being (•qual, the strength is proportional to tlie twist. The effect of variation of the amount of fibre along a yarn is not so directly eviclencecl bnt ma:v be deduced fmm the relationship between the emvcs TN against t I ..;iJ for yarns of iclentit,al quality. As it has been shown, these curves are approximately straight \Yithin the limits B>t I v' N>2 and independent of the eounts. ExpressO Fm.I. 400 300 /~ . (e) ~w r 200 I-..< 0...___ I 2 4 (a) 200's; tc) tOO's, point1< marked X, a separate cun·e is not drawn through these poi11ts as it would be indistinguishable from that of the 200's. (b) !75's. (d) 32's. (ci tOO's, curve obtained by over and undertwisting single threads. _ Isolated points ()Ia), •(b), ·~(c), .-:l(d), are the values of the mean TN. Thev Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 WOUld lie Oll t'UT\"eS (corresponding to (C)) determined by twisting single threarls. 186 SOME PHYSICAL QUANTITIES OF MULE YARN5-0XLEY AND PEIRCE usual variations. Combined with the expression (n-a)tZ=c, this shows n that the effe!'t on the breaking load of a variation of 11 is given b,y .;1=ajn, i.e., the strength of a thin portion of the ;yarn i,; proportionately greater than that of the thil"k plaees. 1t i,; this effect, combined with the drawing out of the thicker places of the roving in mule spinning, which makes it practicable to spin a yarn of moderately uniform strength by automatic drafting. These expressions, of course, do not take account uf the quality or irregularitieR of the raw material. (VI.) THE MEASUREMENT OF REGULARITY OF MULE YARNS. A typical photographic test of the regularity of mule yarns, by the measurement of thicknesR under eompreRsion, iR shown beneath the curves of Plate III. This speedy test follows, and indicates, the ;variations of twist and strength along the draw in a remarkable way. It also com pensates, in much the same way as the tensile strengths, for variations of roving, the thin places having a higher twist which gives them a propor tionately higher value both of '' hardness '' and strength. All the curves agree in showing that the periodic spinning effect is most decided for the finer counts. In the 32's, only a very local increase in twist, hanlne>lR anrl tensile Rtrength is found, and the minimum vaineR are not considerably below the mean valueR for the yarn. (VII.) THICKNESS OF A YARN. The thickness of a yarn, even under slight definite tension, is a rather inrlefinito quantity, owing to the nature of the surface whid1 iR broken by stray hairs. The values given for this quantity were found by photo graphy, using a magnification of about 30 diameters. Stray hairs were neglected and the thickness of the apparent solid portion estimated by callipers. This can be done fairly definitely.* The yarn was wound under a constant tension of 2 grams weight in such a way that the photograph " sampled " the yarn every 2fin. The periods were plainly visible at the junction between the maximum and minimum twists (See Plate IV). The length of the period corresponded exactly with that found in the regularity tester (Zoe. cit., pp. 59-87). The values were grouped in pairs and averaged over two or three periods giving the curves shown in Plate III. The diameter also varies greatly with the number of fibres. A high number of fibres involves a low twist and gives a full yarn; a low number involves a hard twist and a thin yarn. The average values are given in Table III, anrl arc, within the limit of error, proportional to .;N. The demity l·l()(j X 10-3, calculated on the mill-countR of 200"s, lOO's ann 32's, is Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 P= . D2 N 0·66, 0·71, 0·67 respectively, the mean value being O·o8. The actual density of the yarn is probably about 0·7, whic·h meunfl that a little less than half the volume is occupied by cellulose. The variation of any one of the quantities represented in Plate III with any other may be qualitatively, and even approximately quantitatively,· deduced from the curves given; the variation with twist from the periodic * The3e values wJre checked by measuring the areas of a known length of the yarns, as photographed, hy means of an Amsler's planimeter. The mean diameters by the two methods are :- Methorl. 200's. tOO's. 32's. Callipers ...... 2·83 X 10-3 .... 3·62 X 10-3 .... 6·53 X 10-3 ins. Planimeter . ... 3·19 .... 3·94 .... 8·01 Mean .... 3·01 .... 3·78 .... 7·27 (a) Cop 50, 200s. (b) Cop 19, t7Ss. (C) C0/J 23, lOOs. 40 ,, . /\ ,' ' ... ~~--="':.....-/~.~------~--,~ ..... -- ___ .-:..:~-..-:""~- -r.- -!·_-r:_... :Jf/... ___ \,._L_ Twist (tl, ·-·-· " \,,·', ,' ',ll, --..... -· ' .... ' ...... , •' ~. . ' . '/ ./\/'·-·-·-· . Fibre No. (n), · --- · ·:. :·..::::. .. ::."; .. L-.:.:-.=...... ,_ ...... , •.: :~:_ ~·---·::: ...-- - :::.:-.:-r--=~---- -;_.. .._ ...... ,. "' .... •' 60 5 l'ERIODS 8 PERIODS 7 P£RIOD5 10 /'ERIODS 5- 30 00 52. (n-a). t2, x in (inches)2 X XX Breaking length (T .N), 0, in ozs. x 1~ hank. :::O:o 0 G X ~TN. ~ • 0 .. 16 PERIODS 2.9 PERIOJJS e $ a I e G e a reA Diameter (D), A from 0 41 PER/OJ)S microphotograph. X 9-YI Inches X I0-3. 0 Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 .3 PERIODS 0 6-e ~ ' 0 0 \o ~ PERIODS Diameter (H) under comPression from regularity tester. 0·9 Inches x 10-3. 1·5- Single period. 34.- 1·2.- A .a ~ 0 o· PLATE IV. 200's. Hard ------· <-·------Soft tOO's. Hard ---Soft- 32's. Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 Hard Soft 188 SOME PHYSICAL QUANTITIES OF MULE YARNS-QXLEY AND PEIRCE curves, and the variation with the amount of fibre or counts, by comparison of the curves for different yarns. (VIII.) DOUBLED YARNS. In this and the previous work on regularity, the periodicity of single mule-spun yarns has been established. Such yarns, when doubled, will possess irregular doubling twist and tensile strength, depending upon the individual irregularities of the elements of the single yarns which happen to lie side by side. If two thin elements of high twist (corresponding to the peaks of the photographs or those of the tensile strength curves) lie side by side in the doubled yarn, the doubling twist will be abnormally high and the yarn thin and abnormally strong. If two soft twisted elements of the single yarns lie side by side, the doubled yarn will be unusually full, and doubling twist low and the yarn weak. If high and low twist elements of the two single yarns lie side by side, the process of doubling will tend to give a gimp or slub effect, the resulting doubled yarn being of medium strength. In general, the operation of doubling will tend to mask the regular periodicity of the single yarns, for the singles will be laid side by side at random; but where, over a considerable stretch, the hard twisted elements of the two singles are adjacent, the irregularity of the doubled yarn will be pronounced. The undesirability of these properties, especially in knitting yarns, has been emphasised in the previous paper (l·oc. cit., pp. 97-98). The irregularity of the doubled yarn could no doubt be obviated to some extent by doubling single yarns ~vhich have been spun with slightly different mule draws. If at one point two hard elements of the singles happened to lie side by side, the different lengths of the draw would ensure that this coincidence was not repeated except over a length of the doubled yarn which is considerable compared with the length of draw of each of the singles. Alternatively, the same advantage would be secured if the single yarns had equal draws, providing the elements of high twist, or the elements fr·om the points of the cops, are separated initially by half the length of a draw. By adopting one of these devices where doubled mule yarns are w!led, the number of faults (which are accentuated by the subsequent finishing processes applied to the woven or knitted fabrics) would be reduced to a minimum without any modification of the mule mechanism spinning the singles. A number of the routine computations involved in this work have been made by Mr. H. Wardle. SUMMARY The relationship between the local twist and number o£ fibres in a cross-section of a yarn is expressed by a formula with which the twists and Downloaded by [University of Cincinnati Libraries] at 21:32 21 March 2016 fibre numbers of a large number of specimens are in statistical agreement. This relation is shown graphically in Plate II. The formula is generalised by taking account of the fibre cross-section, the mean value of the latter being obtained from the counts and the average number of fibres per section. The constants of the formula vary in a definite manner from count to count, and there appears to be a close correspondence with the mill-rule, viz., twist varies as ¥counts. Knowing the effect of variations in fibre number, it is then possible oo isolate the periodic variation of twist caused by the intermittent action of the mule mechanism. The accompanying variations in strength, visual diameter and " hardness " have been determined and are shown graphically in Plate III. Suggestions are made to minimise or eliminate any effect due to these periodic variations when mule yarns are doubled. [Received for publication 4/8/22.